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Question: If k\(\begin{bmatrix} - 1 & 2 & 2 \\ 2 & - 1 & 2 \\ 2 & 2 & - 1 \end{bmatrix}\)is an orthogonal mat...

If k$\begin{bmatrix}

  • 1 & 2 & 2 \ 2 & - 1 & 2 \ 2 & 2 & - 1 \end{bmatrix}$is an orthogonal matrix then k is equal to-
A

1

B

½

C

1/3

D

None of these

Answer

1/3

Explanation

Solution

Let A = k$\begin{bmatrix}

  • 1 & 2 & 2 \ 2 & - 1 & 2 \ 2 & 2 & - 1 \end{bmatrix}A<sup>T</sup>=k\\ A<sup>T</sup> = k \begin{bmatrix}
  • 1 & 2 & 2 \ 2 & - 1 & 2 \ 2 & 2 & - 1 \end{bmatrix}$

Since A is orthogonal \ AAT = I

̃ k2 $\begin{bmatrix}

  • 1 & 2 & 2 \ 2 & - 1 & 2 \ 2 & 2 & - 1 \end{bmatrix} \begin{bmatrix}
  • 1 & 2 & 2 \ 2 & - 1 & 2 \ 2 & 2 & - 1 \end{bmatrix}$= I

̃ k2[900090009]\begin{bmatrix} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{bmatrix}= Ĩ 9k2 I = I ̃ k2 = 1/9 ̃ k = ±1/3